Theorem cshw1 13568

Description: If cyclically shifting a word by 1 position results in the word itself, the word is build of identical symbols. Remark: also "valid" for an empty word! (Contributed by AV, 13-May-2018.) (Revised by AV, 7-Jun-2018.) (Proof shortened by AV, 1-Nov-2018.)

Assertion

Ref	Expression
cshw1	|- ( ( W e. Word V /\ ( W cyclShift 1 ) = W ) -> A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) )

Distinct variable groups:   i, V   i, W

Proof of Theorem cshw1

Step	Hyp	Ref	Expression
1		ral0 4076	. . . 4 |- A. i e. (/) ( W ` i ) = ( W ` 0 )
2		oveq2 6658	. . . . . 6 |- ( ( # ` W ) = 0 -> ( 0..^ ( # ` W ) ) = ( 0..^ 0 ) )
3		fzo0 12492	. . . . . 6 |- ( 0..^ 0 ) = (/)
4	2, 3	syl6eq 2672	. . . . 5 |- ( ( # ` W ) = 0 -> ( 0..^ ( # ` W ) ) = (/) )
5	4	raleqdv 3144	. . . 4 |- ( ( # ` W ) = 0 -> ( A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) <-> A. i e. (/) ( W ` i ) = ( W ` 0 ) ) )
6	1, 5	mpbiri 248	. . 3 |- ( ( # ` W ) = 0 -> A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) )
7	6	a1d 25	. 2 |- ( ( # ` W ) = 0 -> ( ( W e. Word V /\ ( W cyclShift 1 ) = W ) -> A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) ) )
8		simprl 794	. . . . . . . 8 |- ( ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) /\ ( W e. Word V /\ ( W cyclShift 1 ) = W ) ) -> W e. Word V )
9		lencl 13324	. . . . . . . . . . 11 |- ( W e. Word V -> ( # ` W ) e. NN0 )
10		1nn0 11308	. . . . . . . . . . . . . 14 |- 1 e. NN0
11	10	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( # ` W ) e. NN0 /\ ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) ) -> 1 e. NN0 )
12		df-ne 2795	. . . . . . . . . . . . . . . 16 |- ( ( # ` W ) =/= 0 <-> -. ( # ` W ) = 0 )
13		elnnne0 11306	. . . . . . . . . . . . . . . . 17 |- ( ( # ` W ) e. NN <-> ( ( # ` W ) e. NN0 /\ ( # ` W ) =/= 0 ) )
14	13	simplbi2com 657	. . . . . . . . . . . . . . . 16 |- ( ( # ` W ) =/= 0 -> ( ( # ` W ) e. NN0 -> ( # ` W ) e. NN ) )
15	12, 14	sylbir 225	. . . . . . . . . . . . . . 15 |- ( -. ( # ` W ) = 0 -> ( ( # ` W ) e. NN0 -> ( # ` W ) e. NN ) )
16	15	adantr 481	. . . . . . . . . . . . . 14 |- ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) -> ( ( # ` W ) e. NN0 -> ( # ` W ) e. NN ) )
17	16	impcom 446	. . . . . . . . . . . . 13 |- ( ( ( # ` W ) e. NN0 /\ ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) ) -> ( # ` W ) e. NN )
18		df-ne 2795	. . . . . . . . . . . . . . . 16 |- ( ( # ` W ) =/= 1 <-> -. ( # ` W ) = 1 )
19	18	biimpri 218	. . . . . . . . . . . . . . 15 |- ( -. ( # ` W ) = 1 -> ( # ` W ) =/= 1 )
20	19	ad2antll 765	. . . . . . . . . . . . . 14 |- ( ( ( # ` W ) e. NN0 /\ ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) ) -> ( # ` W ) =/= 1 )
21		nngt1ne1 11047	. . . . . . . . . . . . . . 15 |- ( ( # ` W ) e. NN -> ( 1 < ( # ` W ) <-> ( # ` W ) =/= 1 ) )
22	17, 21	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( # ` W ) e. NN0 /\ ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) ) -> ( 1 < ( # ` W ) <-> ( # ` W ) =/= 1 ) )
23	20, 22	mpbird 247	. . . . . . . . . . . . 13 |- ( ( ( # ` W ) e. NN0 /\ ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) ) -> 1 < ( # ` W ) )
24		elfzo0 12508	. . . . . . . . . . . . 13 |- ( 1 e. ( 0..^ ( # ` W ) ) <-> ( 1 e. NN0 /\ ( # ` W ) e. NN /\ 1 < ( # ` W ) ) )
25	11, 17, 23, 24	syl3anbrc 1246	. . . . . . . . . . . 12 |- ( ( ( # ` W ) e. NN0 /\ ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) ) -> 1 e. ( 0..^ ( # ` W ) ) )
26	25	ex 450	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN0 -> ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) -> 1 e. ( 0..^ ( # ` W ) ) ) )
27	9, 26	syl 17	. . . . . . . . . 10 |- ( W e. Word V -> ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) -> 1 e. ( 0..^ ( # ` W ) ) ) )
28	27	adantr 481	. . . . . . . . 9 |- ( ( W e. Word V /\ ( W cyclShift 1 ) = W ) -> ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) -> 1 e. ( 0..^ ( # ` W ) ) ) )
29	28	impcom 446	. . . . . . . 8 |- ( ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) /\ ( W e. Word V /\ ( W cyclShift 1 ) = W ) ) -> 1 e. ( 0..^ ( # ` W ) ) )
30		simprr 796	. . . . . . . 8 |- ( ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) /\ ( W e. Word V /\ ( W cyclShift 1 ) = W ) ) -> ( W cyclShift 1 ) = W )
31		lbfzo0 12507	. . . . . . . . . . . . . . . . 17 |- ( 0 e. ( 0..^ ( # ` W ) ) <-> ( # ` W ) e. NN )
32	31	biimpri 218	. . . . . . . . . . . . . . . 16 |- ( ( # ` W ) e. NN -> 0 e. ( 0..^ ( # ` W ) ) )
33	13, 32	sylbir 225	. . . . . . . . . . . . . . 15 |- ( ( ( # ` W ) e. NN0 /\ ( # ` W ) =/= 0 ) -> 0 e. ( 0..^ ( # ` W ) ) )
34	33	ex 450	. . . . . . . . . . . . . 14 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) =/= 0 -> 0 e. ( 0..^ ( # ` W ) ) ) )
35	12, 34	syl5bir 233	. . . . . . . . . . . . 13 |- ( ( # ` W ) e. NN0 -> ( -. ( # ` W ) = 0 -> 0 e. ( 0..^ ( # ` W ) ) ) )
36	9, 35	syl 17	. . . . . . . . . . . 12 |- ( W e. Word V -> ( -. ( # ` W ) = 0 -> 0 e. ( 0..^ ( # ` W ) ) ) )
37	36	adantr 481	. . . . . . . . . . 11 |- ( ( W e. Word V /\ ( W cyclShift 1 ) = W ) -> ( -. ( # ` W ) = 0 -> 0 e. ( 0..^ ( # ` W ) ) ) )
38	37	com12 32	. . . . . . . . . 10 |- ( -. ( # ` W ) = 0 -> ( ( W e. Word V /\ ( W cyclShift 1 ) = W ) -> 0 e. ( 0..^ ( # ` W ) ) ) )
39	38	adantr 481	. . . . . . . . 9 |- ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) -> ( ( W e. Word V /\ ( W cyclShift 1 ) = W ) -> 0 e. ( 0..^ ( # ` W ) ) ) )
40	39	imp 445	. . . . . . . 8 |- ( ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) /\ ( W e. Word V /\ ( W cyclShift 1 ) = W ) ) -> 0 e. ( 0..^ ( # ` W ) ) )
41		elfzoelz 12470	. . . . . . . . . 10 |- ( 1 e. ( 0..^ ( # ` W ) ) -> 1 e. ZZ )
42		cshweqrep 13567	. . . . . . . . . 10 |- ( ( W e. Word V /\ 1 e. ZZ ) -> ( ( ( W cyclShift 1 ) = W /\ 0 e. ( 0..^ ( # ` W ) ) ) -> A. i e. NN0 ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) ) )
43	41, 42	sylan2 491	. . . . . . . . 9 |- ( ( W e. Word V /\ 1 e. ( 0..^ ( # ` W ) ) ) -> ( ( ( W cyclShift 1 ) = W /\ 0 e. ( 0..^ ( # ` W ) ) ) -> A. i e. NN0 ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) ) )
44	43	imp 445	. . . . . . . 8 |- ( ( ( W e. Word V /\ 1 e. ( 0..^ ( # ` W ) ) ) /\ ( ( W cyclShift 1 ) = W /\ 0 e. ( 0..^ ( # ` W ) ) ) ) -> A. i e. NN0 ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) )
45	8, 29, 30, 40, 44	syl22anc 1327	. . . . . . 7 |- ( ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) /\ ( W e. Word V /\ ( W cyclShift 1 ) = W ) ) -> A. i e. NN0 ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) )
46		0nn0 11307	. . . . . . . . 9 |- 0 e. NN0
47		fzossnn0 12499	. . . . . . . . 9 |- ( 0 e. NN0 -> ( 0..^ ( # ` W ) ) C_ NN0 )
48		ssralv 3666	. . . . . . . . 9 |- ( ( 0..^ ( # ` W ) ) C_ NN0 -> ( A. i e. NN0 ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) -> A. i e. ( 0..^ ( # ` W ) ) ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) ) )
49	46, 47, 48	mp2b 10	. . . . . . . 8 |- ( A. i e. NN0 ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) -> A. i e. ( 0..^ ( # ` W ) ) ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) )
50		eqcom 2629	. . . . . . . . . 10 |- ( ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) <-> ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) = ( W ` 0 ) )
51		elfzoelz 12470	. . . . . . . . . . . . . . . 16 |- ( i e. ( 0..^ ( # ` W ) ) -> i e. ZZ )
52		zre 11381	. . . . . . . . . . . . . . . . . . 19 |- ( i e. ZZ -> i e. RR )
53		ax-1rid 10006	. . . . . . . . . . . . . . . . . . 19 |- ( i e. RR -> ( i x. 1 ) = i )
54	52, 53	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( i e. ZZ -> ( i x. 1 ) = i )
55	54	oveq2d 6666	. . . . . . . . . . . . . . . . 17 |- ( i e. ZZ -> ( 0 + ( i x. 1 ) ) = ( 0 + i ) )
56		zcn 11382	. . . . . . . . . . . . . . . . . 18 |- ( i e. ZZ -> i e. CC )
57	56	addid2d 10237	. . . . . . . . . . . . . . . . 17 |- ( i e. ZZ -> ( 0 + i ) = i )
58	55, 57	eqtrd 2656	. . . . . . . . . . . . . . . 16 |- ( i e. ZZ -> ( 0 + ( i x. 1 ) ) = i )
59	51, 58	syl 17	. . . . . . . . . . . . . . 15 |- ( i e. ( 0..^ ( # ` W ) ) -> ( 0 + ( i x. 1 ) ) = i )
60	59	oveq1d 6665	. . . . . . . . . . . . . 14 |- ( i e. ( 0..^ ( # ` W ) ) -> ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) = ( i mod ( # ` W ) ) )
61		zmodidfzoimp 12700	. . . . . . . . . . . . . 14 |- ( i e. ( 0..^ ( # ` W ) ) -> ( i mod ( # ` W ) ) = i )
62	60, 61	eqtrd 2656	. . . . . . . . . . . . 13 |- ( i e. ( 0..^ ( # ` W ) ) -> ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) = i )
63	62	fveq2d 6195	. . . . . . . . . . . 12 |- ( i e. ( 0..^ ( # ` W ) ) -> ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) = ( W ` i ) )
64	63	eqeq1d 2624	. . . . . . . . . . 11 |- ( i e. ( 0..^ ( # ` W ) ) -> ( ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) = ( W ` 0 ) <-> ( W ` i ) = ( W ` 0 ) ) )
65	64	biimpd 219	. . . . . . . . . 10 |- ( i e. ( 0..^ ( # ` W ) ) -> ( ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) = ( W ` 0 ) -> ( W ` i ) = ( W ` 0 ) ) )
66	50, 65	syl5bi 232	. . . . . . . . 9 |- ( i e. ( 0..^ ( # ` W ) ) -> ( ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) -> ( W ` i ) = ( W ` 0 ) ) )
67	66	ralimia 2950	. . . . . . . 8 |- ( A. i e. ( 0..^ ( # ` W ) ) ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) -> A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) )
68	49, 67	syl 17	. . . . . . 7 |- ( A. i e. NN0 ( W ` 0 ) = ( W ` ( ( 0 + ( i x. 1 ) ) mod ( # ` W ) ) ) -> A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) )
69	45, 68	syl 17	. . . . . 6 |- ( ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) /\ ( W e. Word V /\ ( W cyclShift 1 ) = W ) ) -> A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) )
70	69	ex 450	. . . . 5 |- ( ( -. ( # ` W ) = 0 /\ -. ( # ` W ) = 1 ) -> ( ( W e. Word V /\ ( W cyclShift 1 ) = W ) -> A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) ) )
71	70	impancom 456	. . . 4 |- ( ( -. ( # ` W ) = 0 /\ ( W e. Word V /\ ( W cyclShift 1 ) = W ) ) -> ( -. ( # ` W ) = 1 -> A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) ) )
72		eqid 2622	. . . . . 6 |- ( W ` 0 ) = ( W ` 0 )
73		c0ex 10034	. . . . . . 7 |- 0 e. _V
74		fveq2 6191	. . . . . . . 8 |- ( i = 0 -> ( W ` i ) = ( W ` 0 ) )
75	74	eqeq1d 2624	. . . . . . 7 |- ( i = 0 -> ( ( W ` i ) = ( W ` 0 ) <-> ( W ` 0 ) = ( W ` 0 ) ) )
76	73, 75	ralsn 4222	. . . . . 6 |- ( A. i e. { 0 } ( W ` i ) = ( W ` 0 ) <-> ( W ` 0 ) = ( W ` 0 ) )
77	72, 76	mpbir 221	. . . . 5 |- A. i e. { 0 } ( W ` i ) = ( W ` 0 )
78		oveq2 6658	. . . . . . 7 |- ( ( # ` W ) = 1 -> ( 0..^ ( # ` W ) ) = ( 0..^ 1 ) )
79		fzo01 12550	. . . . . . 7 |- ( 0..^ 1 ) = { 0 }
80	78, 79	syl6eq 2672	. . . . . 6 |- ( ( # ` W ) = 1 -> ( 0..^ ( # ` W ) ) = { 0 } )
81	80	raleqdv 3144	. . . . 5 |- ( ( # ` W ) = 1 -> ( A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) <-> A. i e. { 0 } ( W ` i ) = ( W ` 0 ) ) )
82	77, 81	mpbiri 248	. . . 4 |- ( ( # ` W ) = 1 -> A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) )
83	71, 82	pm2.61d2 172	. . 3 |- ( ( -. ( # ` W ) = 0 /\ ( W e. Word V /\ ( W cyclShift 1 ) = W ) ) -> A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) )
84	83	ex 450	. 2 |- ( -. ( # ` W ) = 0 -> ( ( W e. Word V /\ ( W cyclShift 1 ) = W ) -> A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) ) )
85	7, 84	pm2.61i 176	1 |- ( ( W e. Word V /\ ( W cyclShift 1 ) = W ) -> A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( W ` 0 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   NNcn 11020   NN0cn0 11292   ZZcz 11377  ..^cfzo 12465   mod cmo 12668   #chash 13117  Word cword 13291   cyclShift ccsh 13534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303  df-csh 13535

This theorem is referenced by:  cshw1repsw  13569